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Index Mathematics Methodology Index Mathematics Methodology May 2021 S&P Dow Jones Indices: Index Methodology Table of Contents Introduction 4 Different Varieties of Indices 4 The Index Divisor 5 Supporting Documents 5 Capitalization Weighted Indices 6 Definition 6 Adjustments to Share Counts 6 Divisor Adjustments 7 Necessary Divisor Adjustments 8 Capped Market Capitalization Indices 10 Definition 10 Corporate Actions and Index Adjustments 11 Different Capping Methods 11 Non-Market Capitalization Weighted Indices 13 Definition 13 Corporate Actions and Index Adjustments 14 Price Weighted Indices 15 Definition 15 Equal Weighted Indices 16 Definition 16 Modified Equal Weighted Indices 17 Corporate Actions and Index Adjustments 17 Multi-Day Rebalancing 18 Exchange Holidays 18 Freeze Date 19 Total Return Calculations 21 Net Total Return Calculations 22 Franking Credit Adjusted Total Return Indices 23 S&P Dow Jones Indices: Index Mathematics Methodology 1 Currency and Currency Hedged Indices 24 Return Definitions 24 The Hedge Ratio 25 Calculating a Currency-Hedged Index 25 Currency Hedging Outcomes 26 Index Computation 26 Dynamic Hedged Return Indices 28 Currency Hedged Excess Return Indices 29 Quanto Currency Adjusted Index 30 Domestic Currency Return Index Calculation 32 Background 32 Equivalence of DCR and Divisor Calculations 32 DCR Calculation 33 Essential Adjustments 33 Risk Control Indices 34 Dynamic Rebalancing Risk Control Index 36 Capped Equity Weight Change 37 Excess Return Indices 37 Exponentially-Weighted Volatility 38 Exponentially-Weighted Volatility Based on Current Allocations 39 Simple-Weighted Volatility 40 Futures-Based Risk Control Indices 41 Exponentially-Weighted Volatility for Futures-Based Risk Control Indices 42 Dynamic Volatility Risk Control Indices 42 Variance Based Risk Control Indices 42 Risk Control 2.0 Indices 43 Constituent Weighting 43 Risk Control 2.0 with Minimum Variance 45 Equity with Futures Leverage Risk Control Indices 46 Weighted Return Indices 47 S&P Dow Jones Indices: Index Mathematics Methodology 2 Leveraged and Inverse Indices 49 Leveraged Indices for Equities 49 Leveraged Indices without Borrowing Costs for Equities 50 Inverse Indices for Equities 50 Inverse Indices without Borrowing Costs for Equities 51 Leveraged and Inverse Indices for Futures 51 Daily Rebalanced Leverage or Inverse Futures Indices 51 Periodically Rebalanced Leverage or Inverse Futures Indices 52 Fee Indices/Decrement Indices 53 Capped Return Indices 56 Dividend Point Indices 57 Alternative Pricing 58 Special Opening Quotation (SOQ) 58 Fair Value Indices 59 Volume-Weighted Average Price (VWAP) 59 Time-Weighted Average Price (TWAP) 59 Negative/Zero Index Levels 60 Index Turnover 61 End-of-Month Global Fundamental Data 62 Monthly Files 62 About the Data 62 Output Files 63 Fundamental Data Points 63 Calculations 64 S&P Dow Jones Indices’ Contact Information 68 Client Services 68 Disclaimer 69 S&P Dow Jones Indices: Index Mathematics Methodology 3 Introduction This document covers the mathematics of equity index calculations and assumes some acquaintance with mathematical notation and simple operations. The calculations are presented principally as equations, which have largely been excluded from the individual index methodologies, with examples or tables of results to demonstrate the calculations. Different Varieties of Indices S&P Dow Jones Indices’ index calculation and corporate action treatments vary according to the categorization of the indices. At a broad level, indices are defined into two categorizations; Market Capitalization Weighted and Non-Market Capitalization Weighted Indices. A majority of S&P Dow Jones Indices’ equity indices are market capitalization weighted and float- adjusted, where each stock’s weight in the index is proportional to its float-adjusted market value. S&P Dow Jones Indices also offers capped versions of a market capitalization weighted index where single index constituents or defined groups of index constituents, such as sector or geographical groups, are confined to a maximum weight. Non-market capitalization weighted indices include those that are not weighted by float-adjusted market capitalization and generally are not affected by notional market capitalization changes resulting from corporate events. Examples include indices that apply equal weighting, factor weighting such as dividend yield or volatility, strategic tilts, thematic weighting or other alternative weighting schemes. S&P Dow Jones Indices offers a variety indices and index attribute data calculated according to various methodologies which are covered in this document: • Market Capitalization Indices: o Market-capitalization indices – where constituent weights are determined by float- adjusted market capitalization. o Capped market-capitalization indices − where single index constituents or defined groups of index constituents, such as sector or geographical groups, are confined to a maximum index weight. • Non-Market Capitalization Indices: o Price weighted indices − where constituent weights are determined solely by the prices of the constituent stocks in the index. o Equal weighted indices − where each stock is weighted equally in the index. • Derived Indices: o Total return indices − index level reflect both movements in stock prices and the reinvestment of dividend income. o Leveraged and inverse indices − which return positive or negative multiples of their respective underlying indices. o Weighted return indices − commonly known as index of indices, where each underlying index is a component with an assigned weight to calculate the overall index of indices level. o Indices that operate on an index as a whole rather than on the individual stocks − these include calculations of various total return methodologies and index fundamentals. S&P Dow Jones Indices: Index Mathematics Methodology 4 o Dividend Point indices − which track the total dividend payments of index constituents. o Risk control, excess return, currency, currency hedged, domestic currency return, special opening quotation, turnover and fundamental data calculations. The Index Divisor The purpose of the index divisor is to maintain the continuity of an index level following the implementation of corporate actions, index rebalancing events, or other non-market driven actions. The simplest capitalization weighted index can be thought of as a portfolio consisting of all available shares of the stocks in the index. While one might track this portfolio’s value in dollar terms, it would probably be an unwieldy number – for example, the S&P 500 float-adjusted market value is a figure in the trillions of dollars. Rather than deal with ten or more digits, the figure is scaled to a more easily handled number (e.g. 2000). Dividing the portfolio market value by a factor, usually called the divisor, does the scaling. An index is not exactly the same as a portfolio. For instance, when a stock is added to or deleted from an index, the index level should not jump up or drop down; while a portfolio’s value would usually change as stocks are swapped in and out. To assure that the index’s value, or level, does not change when stocks are added or deleted, the divisor is adjusted to offset the change in market value of the index. Thus, the divisor plays a critical role in the index’s ability to provide a continuous measure of market valuation when faced with changes to the stocks included in the index. In a similar manner, some corporate actions that cause changes in the market value of the stocks in an index should not be reflected in the index level. Adjustments are made to the divisor to eliminate the impact of these corporate actions on the index value. Supporting Documents This methodology is meant to be read in conjunction with supporting documents providing greater detail with respect to the policies, procedures and calculations described herein. References throughout the methodology direct the reader to the relevant supporting document for further information on a specific topic. The list of the main supplemental documents for this methodology and the hyperlinks to those documents is as follows: Supporting Document URL S&P Dow Jones Indices’ Equity Indices Policies & Equity Indices Policies & Practices Practices Methodology S&P Dow Jones Indices’ Float Adjustment Float Adjustment Methodology Methodology S&P Dow Jones Indices: Index Mathematics Methodology 5 Capitalization Weighted Indices Many of S&P Dow Jones Indices’ equity indices are capitalization-weighted indices. Sometimes these are called value-weighted or market cap weighted instead of capitalization weighted. Examples include the S&P 500, the S&P Global 1200 and the S&P BMI indices. In the discussion below most of the examples refer to the S&P 500 but apply equally to a long list of S&P Dow Jones Indices’ cap-weighted indices. Definition The formula to calculate the S&P 500 is: Pi Qi Index Level = i (1) Divisor The numerator on the right hand side is the price of each stock in the index multiplied by the number of shares used in the index calculation. This is summed across all the stocks in the index. The denominator is the divisor. If the sum in the numerator is US$ 20 trillion and the divisor is US$ 10 billion, the index level would be 2000. This index formula is sometimes called a “base-weighted aggregative” method.1 The formula is created by a modification of a LasPeyres index, which uses base period quantities (share counts) to calculate the price change. A LasPeyres index would be: Pi,1 Qi,o Index = i (2) Pi,0 Qi,o
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